# Thermal conductivity of Si NW

Dear all

I was trying to calculate the thermal conductivity of Si nanowire with Radius 75A at T=300K but I am getting very small value, 0.027[W/mK]. It should some how be around 100 [W/mK].

units real
variable T equal 300
variable V equal vol
variable dt equal 4.0
variable p equal 200 # correlation length
variable s equal 5 # sample interval
variable d equal \$p*\$s # dump interval

# convert from LAMMPS real units to SI

variable kB equal 1.3806504e-23 # [J/K] Boltzmann
variable kCal2J equal 4186.0/6.02214e23
variable A2m equal 1.0e-10
variable fs2s equal 1.0e-15
variable convert equal {kCal2J}*{kCal2J}/{fs2s}/{A2m}

# setup problem

dimension 3
boundary p p p
atom_style atomic

lattice diamond 5.43
region whole block 0 200 0 200 0 100 units box
create_box 1 whole
region wire cylinder z 100 100 75 INF INF units box

lattice hcp 5.43 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
create_atoms 1 region wire

mass 1 28.8
pair_style tersoff
pair_coeff * * …/potentials/Si.tersoff Si
timestep \${dt}
thermo \$d

# equilibration and thermalization

velocity all create \$T 102486 mom yes rot yes dist gaussian
fix NVT all nvt temp \$T \$T 10 drag 0.2

dump 1 all custom 1000 Sigeometry.* id type x y z
run 10000

shell mkdir Silicon
shell cd Silicon

# thermal conductivity calculation

unfix NVT
fix NVE all nve

reset_timestep 0
compute myKE all ke/atom
compute myPE all pe/atom
compute myStress all stress/atom virial

compute flux all heat/flux myKE myPE myStress
variable Jx equal c_flux[1]/vol
variable Jy equal c_flux[2]/vol
variable Jz equal c_flux[3]/vol

fix FLX all ave/correlate \$s \$p \$d &
c_flux[1] c_flux[2] c_flux[3] type auto file Silicon.dat ave running

variable scale equal {convert}/{kB}/\$T/\$T/\$V*s*{dt}

variable k11 equal trap(f_FLX[3]){scale} variable k22 equal trap(f_FLX[4])*{scale}
variable k33 equal trap(f_FLX[5])
\${scale}
thermo_style custom step temp v_Jx v_Jy v_Jz v_k11 v_k22 v_k33
run 10000

variable k equal (v_k11+v_k22+v_k33)/3.0

print "average conductivity: \$k[W/mK] @ \$T K "

Thaks

The thermal conductivity calculation is a recurrent topic in this list. Do some digging into the archives to enlighten yourself a bit while avoiding re-posting identical Qs to others previously answered .

Carlos

Determining thermal conductivity by Green-Kubo has been discussed a lot on the mailing list, you should have a look through the old threads.

The main thing is to check your autocorrelation function, though. Does it decay well to zero? Is the upper limit on your integration appropriate?

As Carlos and Niall emphasize, thermal conductivity is a topic of a lot of posts in this list, so please look through the list. As for your specific example, you seem to be averaging in all directions, which is a bit odd for a nanowire! Also your run of 40 ps seems far too short to get reasonable statistics on the autocorrelation functions. And, as I always comment, be very careful of isotope effects: they are large in silicon at low temperatures! With uniform mass, such as you use, the calculated thermal conductivity should be considerably large than experiment, perhaps a factor of 2 at 300 K (and more at lower temperatures).

Why don’t you try bulk silicon and see if you can get reasonable results before you graduate to nanostructures?

Here are some results that I have exploring the size of the supercell using the Green-Kubo methods for pure Si and Ge, plus some select Si-Ge alloys, using isotopic mixtures of the elements using the pcff+ forcefield

Silicon Germanium ~Si43Ge (2.3%) Si43Ge5(10.4%) 25%
N Atoms Supercell Lambda (W/m.k) Lambda (W/m.K) Lambda (W/m.K) Lambda (W/m.K) Lambda (W/m.K)

64 2x2x2 40.9 12.7 9.5 (3.1%)
512 4x4x4 102.3 29.3 8.5 (2.4%)
4096 8x8x8 153.2 33.8 10.2 (2.3%) 2.6 (10.4%) 1.9
32768 16x16x16 170 52.4

experiment 130 58

I don’t remember off the top of my head the length of the runs – probably about 5 ns – and I take autocorrelation functions at a variety of sampling rates and merge them together to get a single autocorrelation function sampling both short and long times. As you can see, with the Green-Kubo method there is a noticeable dependence on the size of the simulation cell.

I presume this will also occur your nanowire, so you may find a strong dependence with the length of the wire (and diameter, but that is physically meaningful and what your are investigating). Of course the thermal conductivity will probably also be strongly affected by exactly how you terminate the edges of your wire – do you e.g. terminate with hydrogens, allow reconstruction, etc.

Silicon is a challenging case for thermal conductivity (only diamond is more difficult), so while it appears simple it is actually rather challenging.

Paul

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